Well, I incidentally managed to sort this out.
At least for my system. As there may be more than one reason for having no sound with flash.
If it can help you :

1/ The pulseaudio way
(I write this like that because... of another thread around... )
Definitely wanting to get rid of this thing, I dug inside Alsa to find another way.

2/ I read that flash (in absence of pulseaudio) will output sound on the...
Default alsa device !
Having had sound immediately under KDE4, I had not taken care to configure anything about this.
The solution consists in defining this default alsa device, depending on your choice, either as a user preference in $HOME/.asoundrc or system-wide in /etc/asound.conf
Follows the code for my .asoundrc :

This of course to be adapted to your system. I personally get 2 sound cards.
BTW, regarding what we were looking for, only the pcm.!default line together with the descritpion associated with the label are important.
You can get help for the parameters following the card keyword looking in /proc/asound.
Take the name of symlinks to cardX !

but if the browser is running - it is occupying device. nothing else could be played.

You are absolutely right.
In this mode, flash captures the device !

As far as I understand flash capabilities, if this is a problem for you, I think that the only solution consists in passing through some sound daemon.
And, as far as I can see, the only sound daemon flash seems compatible with is... pulseaudio !
You will loose some audio quality and pay an extra 10% cpu charge (if not more) for this...

EDIT : Well, I could be wrong about propulsing pulseaudio as the only solution to this (I agree annoying) problem of device sharing...
there should logically be at least one other through dmix.

Well, as, at the end of the day, you cannot go far without a sound server somewhere, you can solve your problem that way.
I managed to do this with Jack. (without pulseaudio)
If you are still looking for a solution and following this thread, I can provide details.

QJackctl should show a connection established from a Readable Clients / Output ports of the kind alsa-jack.jackP.... to writable Clients / System / playback_1 and playback_2 as soon as you launch flash from your browser.

When I have Flash running somewhere, I get no sound elsewhere. WHY? WHY THE HELL IS THIS SO DIFFICULT? I JUST WANT SOUND TO WORK. IT'S 2010, FORCHRISTSAKE - I COULD PLAY MULTIPLE SOUNDS ON MY ST IN 1985!

[edit] a bit more thinking, as pulseaudio is supposed to know how to merge several sources, i assume it's flash who is trying to use the device directly.... so how do I tell flash to use pulseaudio ?_________________"May God stands between you and harm in all the empty places where you must walk" - Babylon 5

"ERR","pulseaudio[4021]:","","Syslog"," alsa-sink.c: We were woken up with POLLOUT set -- however a subsequent snd_pcm_avail() returned 0 or another value < min_avail."
"ERR","pulseaudio[4021]:","","Syslog"," alsa-sink.c: Most likely this is a bug in the ALSA driver 'snd_hda_intel'. Please report this issue to the ALSA developers."
"ERR","pulseaudio[4021]:","","Syslog"," alsa-sink.c: ALSA woke us up to write new data to the device, but there was actually nothing to write!"

You don't say

Quote:

I suggest you try an improved dmix.

So far I could not make any much sense of dmix. I tried some (as below) but none worked so far.

You can do the following things. In terminal write,
>> sudo apt purge flash*
and then run
>> sudo apt-get update && sudo apt-get upgrade
then again run
>> sudo apt-get install adobe-flashplugin
I am extremely new to Linux, so I am not sure if this will work. hopefully it does.